A rigid vessel containing a 3:1 mol ratio of carbon dioxideand water vapor is held at 200 °C where it has a total pressureof 202.7 kPa. If the vessel is cooled to 10 °C so that all of thewater vapor condenses, what is the pressure of carbon dioxide?Neglect the volume of the liquid water that forms on cooling.

Verified step by step guidance

1. Identify the initial conditions: The vessel contains a 3:1 mol ratio of CO2 to H2O vapor at 200 °C with a total pressure of 202.7 kPa.
2. Use the ideal gas law to find the partial pressures of CO2 and H2O at the initial state. Since the total pressure is 202.7 kPa and the mol ratio is 3:1, calculate the partial pressures using the formula: P_total = P_CO2 + P_H2O.
3. Calculate the partial pressure of CO2: Since the mol ratio is 3:1, the partial pressure of CO2 is three times that of H2O. Let P_H2O = x, then P_CO2 = 3x. Use the equation: 3x + x = 202.7 kPa to find x.
4. Determine the partial pressure of CO2 at the initial state: Substitute the value of x back into P_CO2 = 3x to find the initial partial pressure of CO2.
5. Since the vessel is cooled to 10 °C and all water vapor condenses, the pressure of CO2 remains the same as its initial partial pressure because the volume of the vessel and the amount of CO2 remain constant.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding how gases behave under varying conditions, particularly in rigid vessels where volume remains constant. In this scenario, it helps to calculate the pressure changes of the gases as temperature changes.

Partial Pressure

Partial pressure is the pressure exerted by a single component of a gas mixture. According to Dalton's Law, the total pressure of a gas mixture is the sum of the partial pressures of its individual components. This concept is crucial for determining the pressure of carbon dioxide after the water vapor condenses, as it allows us to isolate the contribution of carbon dioxide to the total pressure.

Phase Change and Vapor Pressure

Phase change refers to the transition of a substance from one state of matter to another, such as from gas to liquid. Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid at a given temperature. Understanding these concepts is vital for this problem, as cooling the vessel causes water vapor to condense, affecting the overall pressure and the remaining gas components.